We study the duration and variability of late-time X-ray flares following gamma-ray bursts (GRBs) observed by the narrow-field X-ray telescope (XRT) aboard the Swift spacecraft. These flares are thought to be indicative of late-time activity by the central engine that powers the GRB and produced by means similar to those which produce the prompt emission. We use a nonparametric procedure to study the overall temporal properties of the flares and a structure function analysis to look for an evolution of the fundamental variability timescale between the prompt and late-time emission. We find a strong correlation in 28 individual X-ray flares in 18 separate GRBs between the flare duration and their time of peak flux since the GRB trigger. We also find a qualitative trend of decreasing variability as a function of time since trigger, with a characteristic minimum variability timescale Δt/t = 0.1 for most flares. The correlation between pulse width and time is consistent with the effects of internal shocks at ever-increasing collision radii, but could also arise from delayed activity by the central source. Contemporaneous detections of high-energy emission by GLAST could test between these two scenarios, as any late-time X-ray emission would undergo inverse Compton scattering as it passes through the external shock. The profile of this high-energy component should depend on the distance between the emitting region and the external shock.